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Chi Square Random Variable

Posted By Krishna Sankar On July 28, 2008 @ 6:14 am In DSP | 12 Comments

While trying to derive the theoretical bit error rate (BER) for BPSK modulation in a Rayleigh fading channel, I realized that I need to discuss chi square random variable prior.

What is chi-square random variable?

Let there be independent and identically distributed Gaussian random variables with mean 0 and variance and we form a new random variable,

.

Then is a chi square random variable with degrees of freedom.

There are two types of chi square distribution. The first is obtained when has a zero mean and is called central chi square distribution. The second is obtained when has a non-zero mean and is called non-central chi square distribution. Four our discussion, we will focus only on central chi square distribution.

PDF of chi-square random variable with one degree of freedom

Using the text in Chapter 2 of [DIGITAL-COMMUNICATION: PROAKIS] [1] as reference.

The most simple example of a chi square random variable is

,

where
is a Gaussian random variable with zero mean and variance .

The PDF of is
.

By definition, the cumulative distribution function [2](CDF) of is
.

This simplifies to

.

Differentiating the above equation with respect to to find the probability density function,

.

Summarizing, the pdf of chi square random variable with one degree of freedom is,

.

PDF of chi-square random variable with two degrees of freedom

Chi square random variable with 2 degrees of freedom is,

,

where,
and are independent Gaussian random variables with zero mean and variance .

In the post on Rayleigh random variable, we have shown that PDF of the random variable,

where is

.

For our current analysis, we know that

.

Differentiating both sides,

.

Applying this to the above equation, pdf of chi square random variable with two degrees of freedom is,
.

PDF of chi-square random variable with m degrees of freedom

The probability density function is,

, where

the Gamma function is defined as,

,

p an integer > 0

.

I do not know the proof for deriving the above equation. If any one of you know of good references, kindly let me know. Thanks. :)

Simulation Model

Just for your reference, Matlab/Octave simulation model performing the following is provided

(a) Generate chi square random variables having m=1, 2, 3, 4, 5 degrees of freedom

(b) Probability density function is computed and plotted

Click here to download: Matlab/Octave script for simulating PDF of chi square random variable [3]

PDF of chi square random variable

Figure: PDF of chi square random variable (=1)

Reference

[DIGITAL-COMMUNICATION: PROAKIS] [1] Digital Communications, by John Proakis [1]


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URLs in this post:

[1] [DIGITAL-COMMUNICATION: PROAKIS]: http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FDigital-Communications-John-Proakis%2Fdp%2F0072321113&tag=dl04-20&linkCode=ur2&camp=1789&creative=9325

[2] cumulative distribution function : http://en.wikipedia.org/wiki/Cumulative_distribution_function

[3] Matlab/Octave script for simulating PDF of chi square random variable: http://www.dsplog.com/db-install/wp-content/uploads/2008/07/pdf_chi_square_random_variable.m

[4] click here to SUBSCRIBE : http://www.feedburner.com/fb/a/emailverifySubmit?feedId=1348583&loc=en_US

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